Application of the unified method to solve the ion sound and Langmuir waves model

We present the unified method and use it to integrate the ion sound and Langmuir waves (ISLW) model to retrieve optical soliton solutions. Some new dynamical optical solitons involving the combo of rational, trigonometric, and hyperbolic function solutions are added in this study. The derived optical soliton solutions display various properties such as beat pattern and oscillation with increasing, decreasing, and simultaneously increasing and decreasing amplitudes. Moreover, kink, dark bell, singular kink, single breather, multiple breathers, dark-, bright-, and dark bright periodic waves are founded. Finally, some dynamical characteristics of the acquired solutions are depicted.


Introduction
Optical solitons are one of the rising research areas for the development of the telecommunication industry. We cannot imagine the operating of optical fiber, email, internet, mobile phones and many other communications except the idea of a solitary wave [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. In recent years, nonlocal integrable NLS [16], [17] and mKdV [18], [19] equations have been systematically analyzed through the Hirota bilinear technique [20] and the Riemann-Hilbert algorithm [21]. With the development of the soliton concept, various nonlinear models like the Dullin-Gottwald-Holm model [22], the first integro-differential KP hierarchy model [23], the geophysical Korteweg-de Vries model [24], the Korteweg-de Vries-Burgers model [25], the Jimbo-Miwa model [26], and others [27], [28], [29], [30], [31] are emerging in the telecommunication industry. A well-known model, named as the ion sound and Langmuir waves models, was first presented in 1972 [32] and has since received numerous accolades. This model was extensively employed in various solitary wave propagation including plasma. This model can describe the dynamical behavior of the ion sound wave, which is caused by a high-frequency field for the impact of the ponderomotive force and the Langmuir wave, which is one kind of nonlinear evolution equation [33], [34]. To  integration algorithms such as inverse scattering [35], trial equation scheme [36], Sine-Gordan expansion scheme [37], modified Kudraysov scheme [38], extended direct algebraic mapping method [39], and others [40], [41] have been studied. The fractional ISLW model is studied in ref. [42], [43], and the fractional generalized HSC KdV model is studied in ref. [44]. The first goal of this manuscript is to acquire optical soliton solutions that clarify the physical structures of the governing model by the unified technique [45], [46], [47]. This technique is a generalization of two well-known schemes, known as tanh-function, and ′ ∕ -expansion, shown by Akkagil and Aydemir [45]. In 2021, Ullah and others found the singular solution to the LPD equation [46]. This method was also used to solve the Biswas-Arshed nonlinear structure in 2022 [47]. We affirm that the optical soliton solution of the ISLW model in this method has not been studied yet.

Governing model
The ISLW model has the following form [33], [34]: plasma frequency, and the parameter denotes the normalized density perturbation. In engineering and applied sciences, especially in high-frequency cases, the ISLW model is applicable for the impact of ponderomotive force.

The ODE structure of the model
Assume the subsequent relation for solving Eq. (1): in which , , , and are real parameters need to be measured afterward. Combining Eq. (2) and Eq. (1) yields Separating imaginary and real parts from the first part of Eq. (3) correspondingly gives From Eq. (4) we get = − . Integrating the 2nd part of Eq. (3) two times with regard to , we have where ≠ ±1. From Eq. (5) and Eq. (6) we have

Summary of the unified method and its application
Suppose the trial solution of Eq. (7) is Eq. (9) contains nine types of solutions in three cases: Case-01: Hyperbolic function (when is negative): Case-02: Trigonometric function (when is positive): Case-03: Rational function (when = 0) when ≠ 0, , and are arbitrary parameters.

Figure analysis
The solutions 11 , 12 , 21 , 22 , 31 , and 32 give similar behavior as depicted in Fig. 1 (a-c) by 21 . The character of this graph changes based on the condition for parameter . We see that Fig. 1 (a) displays oscillations with rapidly growing amplitudes at = −1; Fig. 1 (b) illustrates oscillations with rapidly reducing amplitudes, then drops to 0, and then again oscillations with rapidly growing amplitudes at = −1∕7; and Fig. 1 (c) exhibits oscillations with rapidly reducing amplitudes at = 1. A combination of rational polynomials and periodic solitons produces periodic waves 19 and 39 , as depicted in Fig. 2  (a-d) by 39 . The combo periodic optical soliton solutions 13 , 23 , and 33 exhibit similar behavior and can increase or decrease wave amplitudes. Optical communication systems make use of this property extensively. In particular, we depicted the solution 13 in Fig. 3 (a-c). We see that for > 0, the wave amplitude increases with a constant height after a definite time (see Fig. 3 (a)); for = 0 (see Fig. 3 (b)), it remains the same; and for > 0, it decreases (see Fig. 3 (c)). The solutions 14 , 24 , and 34 represent double periodic optical solutions as depicted in Fig. 4 (a-d) by 14 . This diagram displays oscillations    Fig. 5 (a-f). When = 1, the solution represents a singular wave (see Fig. 5 (a-c)). When = 2, the solution is not a singular wave (see Fig. 5 (d-f)). The solutions 13 , 23 and 33 exhibit kink soliton, as pictured by the solution 13 in Fig. 6.1  (a, b). 14 and 34 exhibit dark bell soliton solution, as pictured by the solution 14 in Fig. 6.2 (a, b). 21 , 22 , 31 , and 32 exhibit singular kink soliton, as pictured by the solution 22 in Fig. 6.3 (a, b). The solutions 11 , 12 , 19 , 24 , and 39 exhibit a single breather wave solution as shown in Fig. 7.1 (a, b) by the solution 11 . The solutions 17 , 18 , 27 , 28 , 37 , and 38 exhibit multiple breather wave solutions as depicted in Fig. 7.2 (a, b) Fig. 8 (a-c). This solution represents a dark periodic wave if 2 < 2 (see Fig. 8 (a)), a dark-bright periodic wave if = ± (see Fig. 8 (b)), and a bright periodic wave if 2 > 2 (see Fig. 8 (c)). We can say that the results acquired for the ISLW model in the unified approach are unprecedented and newest.

Conclusion
This paper has presented the unified method to integrate the ISLW model to retrieve optical solitons that can be used in birefringent fibers. Fig. 2 illustrates periodic waves whose amplitudes increase and decrease. The derived optical soliton solutions exhibit some dynamics as beat pattern, oscillation together increasing, decreasing, and jointly increasing-decreasing amplitudes as shown in Fig. 1, 3-5. Kink, dark bell, and singular kink waves are shown in Fig. 6. Single breather and multiple breather waves are shown in Fig. 7. Moreover, dark, dark-bright, and bright solitons are shown in Fig. 8. In mathematics and engineering, the mentioned technique is an elegant algorithm for finding optical solitons of nonlinear complex models. In addition, applications of fractional differential equations to ISLW models are described in Ref. [42], [43]. In future studies, we need to develop the fractional differential equation of this model by the unified method.